Study of Heavy Crude Oil Flows in Pipelines with Electromagnetic

Article pubs.acs.org/EF

Study of Heavy Crude Oil Flows in Pipelines with Electromagnetic Heaters Ricardo Dunia* and Thomas F. Edgar Department of Chemical Engineering, The University of Texas at Austin, Austin, Texas 78712, United States ABSTRACT: The electromagnetic heating of heavy crude oil in cylindrical pipes represents a novel technique to reduce fluid viscosity and diminish the cost required for its transportation. In this study, the oil viscous fluid momentum and energy balances, which include the effects of electromagnetic heating, variable viscosity, and fluid dielectric properties, are solved in cylindrical coordinates. The electromagnetic energy absorbed by oil is converted into sensible heat, which could significantly reduce the viscosity of heavy crude oil fluids. This drop in fluid viscosity diminishes flow pressure losses in pipelines, which reduces the number and size of pumping stations between oil producers and consumers. Different pipe materials are considered here to determine the effect of their dielectric properties in the fluid flow. Simulation results show that pipe materials with large electromagnetic absorption tend to attenuate the pipe interior electromagnetic field, which reduces the direct warming of the fluid. This significant reduction in the direct fluid heating suggests that pipes made by transparent electromagnetic materials are preferred for this type of application. A fixed cost analysis of using electromagnetic heaters was made to maximize the distance between pump stations in a long pipeline stretch. The simulation results demonstrate that electromagnetic heaters increase the distance between pump stations by 30%.

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INTRODUCTION Heavy crude oil transportation costs have represented a major factor for the price and use of such a fuel at energy consumption facilities.1 Because heavy oil produced hundred of miles from consumers has become an indispensable source of energy, its transportation cost is still considered high compared to other fuels.2 Heavy oil transportation pipelines have been built in Alaska, Canada, Colombia, central Asia, Africa, and California. The published literature refers to five different methods of heavy oil transportation in pipelines.3 Among them, heating is an attractive method for improving the flow properties of heavy crude oils because viscosity decreases very rapidly with an increasing temperature.4 As an example, Figure 1 illustrates the Chad−Cameroon pipeline in central Africa, where six pump−heating stations are used to transport 500 000 barrels per day (BPD) along a 640 mile distance between Palogue and Port Sudan.5 Electromagnetic heating applications are receiving significant attention because of the selectivity in which energy is delivered to electromagnetic absorbing materials.6−9 When energy is transferred to a fluid via electromagnetic waves, the heat of absorption is not limited to the surface but also to inner fluid layers, allowing for the increase of energy transference. The penetration capacity of electromagnetic furnaces provides efficient volumetric heating for industrial applications as an alternative to conventional thermal processing methods.10 This is the case for industrial drying processes, where the energy consumed was reduced by a factor between 2 and 10.11 Furthermore, electromagnetic heating is also ideal for applications that require fast thermal responses.12 In summary, the advantages of using microwaves for industrial processing include fast heat transfer, volumetric and selective heating, compactness of equipment, speed of switching on and off, and minimal environmental impacts. Microwave leakage can © 2012 American Chemical Society

certainly be kept well below regulatory levels, and heating stations can be optimally controlled in an unmanned fashion. The latest technology developments enforce safety considerations in units that do not require human intervention, and the fast transference of electromagnetic energy permits fast control responses to forecast environmental changes. In the case of heavy oil pipelines that require pumping and heating substations, the distribution of electromagnetic heating devices between oil producers and consumers may provide a considerable advantage when compared to fired heating and heavy equipment pump stations. Furthermore, electromagnetic heaters (EHs) can be placed within pump stations to reduce the crude oil viscosity. The simulations presented in this work provide a first step in the estimation of electromagnetic heating for heavy crude oil transportation. In general, electromagnetic heating tends to be energyefficient when the material to be heated is well-defined and can be enclosed inside a furnace with reflective electromagnetic properties. In that way, losses can be minimized and the energy generated by the electromagnetic field source can be readily absorbed by the processing material. This is the case for ceramic furnaces and industrial dryers driven by electromagnetic heating, which show an energy savings that reduces power consumption to a fraction of the amount required when using fuel or steam.11 Electromagnetic waves with a frequency from 300 MHz to 300 GHz correspond to microwave frequency. Industrial microwave frequencies are usually regulated by the policies of each country. In the case of North America, the Federal Communications Commission allows for frequencies of 915, Received: March 29, 2012 Revised: June 16, 2012 Published: June 19, 2012 4426

dx.doi.org/10.1021/ef300537e | Energy Fuels 2012, 26, 4426−4437

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Figure 1. Chad−Cameroon pipeline shows six pumping stations along a 640 mile extension. Each station includes three centrifugal pumps in parallel, heaters, and tankage for fuel storage.

2450, 5800, and 22 000 MHz for industrial use.8 For laboratory use, a frequency of 2450 MHz is preferred because it has adequate penetration depth in solid-state applications.13 Many publications have shown the use of 915 and 2450 MHz electromagnetic heat sources for oil recovery in heavy oil reservoirs.14−17 These studies show that the warming of heavy crude oil in petroleum reservoirs as a substitute of hot water/ steam injection and in situ combustion provides energy savings. The main obstacles for the development of this oil extraction technology are equipment cost, safety consideration because of the existence of hot spots as a consequence of heterogeneous crude oil material, and the need for electric power, which is a costly heating source.18 In this work, the heating of a heavy crude oil with known dielectric properties along a straight pipe heated by an electromagnetic field is studied. The effect of the pipe dielectric properties and amplitude of the magnetic field are taken into account in the design of this EH. Pressure drop and bulk temperature profiles are used to determine the advantages and repercussions of using this electromagnetic heating technology, where the fluid does not need to be withdrawn from the pipe to reduce its viscosity. Nevertheless, the availability of electric power in remote locations may represent a challenge in the development of this technology. The optimal location of EH is also analyzed in this work to maximize the distance between pump stations. In that way, less pump stations will be needed as the viscosity is reduced by increasing the temperature of the crude oil inside the EH. The decrease in viscosity will reduce the pressure drop in the pipeline, allowing for a more effective use of the power supplied by the centrifugal pumps. However, the cost associated with the installation and operation of EH is not considered in the scope of this work. This paper is organized in the following manner. The Economic Motivation section illustrates how EHs can reduce crude oil transportation costs. The Fluid Model Equations section demonstrates the expressions that define the energy and momentum balance for the transported crude oil flow in

cylindrical coordinates. The effect of the temperature in the electromagnetic permittivity and viscosity of the crude oil is provided in the Physical Property Variations section. The Simulation Cases and Results section demonstrates simulation and results for different types of pipe materials. An Economic Study Analysis section of the optimal location and spacing of EH in heavy crude oil transportation is presented before the Conclusion of this research work.

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ECONOMIC MOTIVATION Heating the heavy crude oil and pipelines represents the second most frequently used method for transporting heavy oil in pipelines.2 The principle is to conserve the elevated temperature (around 100 °C) at which the oil is produced through insulation of the pipelines. Nevertheless, sequential heating of warmed heavy oil is needed because of heat losses with the environment and the effect of the temperature on viscosity.19 Therefore, the method of heating heavy oil is economically viable for long pipelines when oil is reheated in the pumping stations and pipelines are properly insulated. Insulation options include an extra layer of insulation and burying the pipeline to conserve heat.3 Special heaters are used along the way to maintain the temperature in the range of 50−80 °C. Nevertheless, the pressure drop can reach 1000 psi over a 40 mile stretch for significant heavy crude oil.20 Because of their high maintenance, these heaters operate in parallel with a backup unit in case of equipment failure or decoking. Fired heaters only heat the fluid in contact with the pipe wall and are prone to an uneven heating distribution that can generate flammable vapors inside the transportation pipes. Therefore, this type of heater requires downstream flash vessels to strip the vapors generated during heating. These vapors can be used as part of the fuel required in the fire heater combustion chamber. Fired heaters are used in the Chad−Cameroon pipeline system of Figure 1, where two heaters per pump station provide 105 000 British thermal units (Btu)/h to raise the pump suction temperature to 80 °C. 4427


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Figure 2. Crude oil transportation diagram and pipe pressure profiles. The top diagram shows the case of no intermediate heating between pump stations. The energy losses because of high viscosity at low temperatures forces a separation of only Y miles between stations. The installation of an EH between pump stations (bottom diagram) allows for an extra separation of ΔY miles between stations, reducing the number of stations and pump head required for significant transportation distances.

These profiles show the effect of EH on the pressure gradient and bulk temperature. This work does not intend to study the feasibility of when EH should be installed for a particular transportation case. However, calculations related to the reduction in pump stations and the optimal location of EH are provided in the Economic Study Analysis section.

Shell and tube heaters (referred as indirect heaters) are also used in pump stations to heat and reduce the viscosity of the heavy oil. Steam is used on the tube side to reduce the crude oil pressure drop along the heat exchanger. Although this type of exchanger is easier to maintain and has a lower risk of nonuniform heating than fired heaters, the fluid pressure drop is significantly larger than fired heaters. The use of EHs along a significant pipeline stretch can be installed between pump stations to reduce the fluid viscosity. This type of configuration reduces the number of pump stations required to transport heavy oil from producers to consumers. The fact that electromagnetic heating is performed inside the same cylindrical pipe size used for flow transportation and that the heating could be radially distributed to the core of the fluid flow makes this technique attractive for installation, maintenance, and operational cost savings. Figure 2 illustrates the use of electromagnetic heating between pump stations for crude oil transportation. The loss of energy to the ambient surroundings decreases the fluid temperature, which increases the heavy oil viscosity. Large viscosities increase the pressure gradient considerably, to a point that a second pump station is required. The use of an EH obviates a second pump station by increasing the temperature of the fluid without bringing the fluid out of its normal course of flow. As shown in Figure 2, an extra pipe distance, denoted by ΔY, is realized for crude oil transportation with an EH before the fluid pressure in increased in pump station B. Ultimately, fewer pump stations will be needed to cover distances of hundreds of miles when EHs are designed and placed appropriately. The advantages of using EH in the place of shell-tube heat exchangers between pump stations are the reduction in pressure losses (the shell-tube heat exchanger pressure drop is significant) and removing the need of a hot-fluid source. The distance between pump stations could be in the order of 50 miles, which may prohibit the use of an alternative source of energy, such as steam or combustible fuel. Electric power can be easily transported in power lines and used for EH while distributing electric power for pumps and instrumentation equipment. Furnace heaters may have low pressure losses but are more expensive to maintain and require accessory equipment that increases their cost of installation. The next section demonstrates the solution of the velocity and temperature profiles of heavy crude oil transported through a cylindrical pipe that undergoes electromagnetic heating.

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FLUID MODEL EQUATIONS

Temperature and pressure profiles along the EH allow for the determination of the size and pipe electromagnetic properties necessary for the optimal design of this equipment. The pipe wall temperature calculations are also necessary to eliminate the risk of heavy fuel ignition and proper distribution of heat in the radial direction. Finally, the stream outlet conditions are important to determine the location of pump stations for heavy oil transportation. This last study is included in the Economic Study Analysis section of this work. The crude oil flow momentum and energy balance equations in cylindrical coordinates are given by

∂ 1 ∂ 1 ∂ ⎛⎜ ∂U ⎞⎟ dP − (ρU 2) + (rρVU ) = rμ ∂x r ∂r r ∂r ⎝ ∂r ⎠ dx

(1)

∂ 1 ∂ 1 ∂ ⎛⎜ ∂T ⎞⎟ + qḟ (ρUCT ) + (rρVCT ) = rk ∂x r ∂r r ∂r ⎝ ∂r ⎠

(2)

where the viscous dissipation and the axial diffusion terms have been neglected. ∂ ⎛⎜ ∂U ⎞⎟ μ ≃0 ∂x ⎝ ∂x ⎠

∂ ⎛⎜ ∂T ⎞⎟ k ≃0 ∂T ⎝ ∂x ⎠

The pressure gradient dP/dx only changes in the axial direction x. The fluid flow is considered laminar (Re < 2300), steady-state (∂/∂t = 0), and incompressible (∂ρ/∂P = 0). The term q̇f refers to the electromagnetic heat rate of absorption per unit volume. In most dielectric materials, the heat absorbed as a result of the magnetic field component is neglected12 and will not be taken into account in this study. The electric field intensity at r = R, denoted by E⃗ R, is considered perpendicular to the pipe surface. Such an assumption permits the substitution of the electric field vector by its amplitude at the pipe wet wall, ER. Therefore, a good approximation of the electromagnetic power absorbed per unit volume of fluid is given by21 qḟ (r ) = 2π -ε0ε″f E R 2e−2αf (R − r),

0≤r≤R

(3)

where ε″f is the effective loss factor and represents the imaginary part of the fluid dielectric permittivity, εf. The permittivity depends upon the dielectric properties of the fluid and electromagnetic field frequency, - . The attenuation of the electromagnetic wave is 4428


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characterized by the electromagnetic absorption coefficient α, given by22 1/2 ⎡ ⎛ ⎞⎤ ⎛ ε″f ⎞2 2π ⎢ 1 ⎜ ⎥ ⎟ αf = ε′f 1 + ⎜ ⎟ − 1⎟⎥ λ ⎢⎢ 2 ⎜ ⎝ ε′f ⎠ ⎝ ⎠⎥⎦ ⎣

For the momentum balance equation 1 ∂ 1 ∂ (ρ*U *2 ) + (ηρ*V *U *) Re ∂z η ∂η ∂U * ⎞ 2 ∂ ⎛ 1 dP * = ⎟− ⎜ημ* Reη ∂η ⎝ Re dz ∂η ⎠

(4)

Table 1 provides the value and description of the parameters required to evaluate q̇f in eq 3. The case of constant fluid dielectric properties

Table 1. Model Parameters ε0 ε′e

vacuum permittivity relative dielectric constant at the entrance amplitude of the electric field at r = R amplitude of the electric field at r = R0 frequency free space wavelength loss factor at the entrance, ε″e/ε′e electromagnetic heat conversion at the entrance viscosity at the entrance density entrance temperature average axial velocity pipe internal radius pipe external radius Reynolds number Prandtl number fluid heat capacity fluid conductivity

ER ER0

λ tan (δe) ε″e μe ρ Te U̅ R R0 Re Pr C kf

For the energy balance equation 1 ∂ 1 ∂ (ρ*U *C*θ ) + (ηρ*V *C*θ ) Re ∂z η ∂η 2 ∂ ⎛ ∂θ ⎞ = ⎜ηk* ⎟ + q*̇ f Peη ∂η ⎝ ∂η ⎠

a

description

value 8.85410 3.9

units

−12

reference

F/m

V/m V/m

2.45 12.2 1.58 × 103

GHz cm

6.16 × 10

3

180 902−940 20 0.25 0.5 0.5025 1230 2250 2000 0.16

1 ∂ ⎛ ∂Ue ⎞ dP ⎜rμ ⎟= r ∂r ⎝ e ∂r ⎠ dx

11 11 23

24 24

2 ∂ ⎛ ∂U *e ⎞ dP * ⎟= ⎜η dz η ∂η ⎝ ∂η ⎠

J kg−1 K−1 W m−1 K−1

The letters “f” and “e” stand for fluid and at the heater entrance conditions, respectively.

and uniform magnetic field provides the following heat rate absorbed by the fluid: (5)

q = 2πRke

where l represents the length of the electromagnetic heating section. Equations 1 and 2 generate a system of partial differential equations that are exclusively parabolic.25 The continuity equation can be expressed in integral form

∫(A) ρU dA = ṁ

∂θ |η = 1 = q*, ∂η

(6)

R

R

(7)

which are important parameters to determine the effectiveness of the electromagnetic heating for crude oil transportation. On the basis of the dimensionless variables and input parameters defined in the Appendix, differential eqs 1, 2, and 6 can be expressed in the following dimensionless form: For the integral continuity equation

∫0

1

ρ*U *ηdη =

1 2

∂θ |η = 0 = 0 ∂η

where q* ≡ q/(2πke(To − Te)). Two different cases for q are considered depending upon the pipe dielectric properties. These are as follows: (a) pipe with transparent electromagnetic properties (in such a case, the fluid is directly heated by the electromagnetic source and the pipe is thermally insulated; i.e., q = 0) and (b) pipe heated by the electromagnetic energy source (in this case, the fluid is warmed by not only direct exposure to the electromagnetic source but also the pipe wall). The Simulation Cases and Results section provides the solution profiles of these two cases.

∫0 ρU (x , r)T(x , r)r dr ∫0 ρU (x , r)r dr

∂T |r = R ∂r

while ∂T/∂r|r=0 = 0 because of symmetry. In terms of dimensionless variables, these boundary conditions result in

where ṁ is a constant and A represents the cross-section area of flow. The solution of eqs 1, 2, and 6 provides the pressure gradient and the average bulk temperature Tb(x) =

(12)

subject to dU*e/dη|η=0 = 0 and U*e(1) = 0. The solution of eqs 8−12 provides U*e(η) = 2(1 − η2) and dP*/dz = −16 for entrance conditions to the electromagnetic heating section. A pressure gradient above −16 indicates a reduction in the pressure drop because of fluid heating. Positive values of dP*/dz suggest that the heater permits the fluid to accelerate inside the pipe, making the heater act as a pump device by increasing the fluid pressure inside the pipe. Such situations rarely occur in practice for laminar flow regimes. The boundary conditions in the axial velocity are identical to the conditions used for the entrance profile. The radial velocity boundary conditions are given by V*(z, 0) = V*(z, 1) = 0. For the energy balance equation, the external boundary condition is based on the external heat flux per unit length q

a

Q̇ f = lπR2qḟ

(11)

Notice that radial velocities are zero for developed flows; i.e., V(r) = 0. The entrance pressure and temperature profiles are considered uniform and known, which indicates P(0, r) = Pe and T(0, r) = Te, respectively. The dimensionless form of eq 11 is given by

23 cP kg/m3 °C m/s m m

(10)

where the dimensionless temperature θ(z, η) and heat q̇*f(η) are also defined in the Appendix. An axial-developed profile is considered for entrance conditions in the momentum balance shown in eq 1. Such a profile is denoted by Ue(r) and is given by the solution of

23

180

(9)

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PHYSICAL PROPERTY VARIATIONS The viscosity and dielectric properties of the heavy crude oil are affected the most by temperature changes. The effect of the temperature on density, specific heat, and conductivity can be considered negligible. Although there are significant reserves of heavy crude oil in Canada and Venezuela, the publication of

(8) 4429


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Table 2. Dielectric Permittivity for the Russkoye Oil23

heavy oil dielectric properties have been limited in the literature. The Russkoye crude oil produced in Russia has the most complete published dielectric properties and is used in this research. The Russkoye oil has an American Petroleum Institute (API) gravity between 20° and 25°, and its transportation temperature is between 20 and 100 °C. Although a crude oil with an API gravity in the low twenties is considered in the limit between medium and heavy oil, the Russkoye oil has been considered heavy crude oil in previous publications.23 Viscosity. The correlation developed by Bennison26 for heavy crude oil viscosities has demonstrated accurate results that match actual values and is used in the present work. The correlation follows the form:

where h( ) and g( ) represent the following polynomial functions of the crude oil API gravity: h(x) = −0.8021x + 23.8765 g (x) = 0.31458x − 9.21592

This correlation gives good results for crude oil with an API gravity even greater than 20 and temperatures above 120 °C. Dielectric Properties. As mentioned above, there is a lack of electromagnetic property information in the literature for crude oil. Among the few published dielectric properties for heavy oils are the ones for southeast Turkey crude oil,27 Russkoye crude oil field in Russia,23 Alaska heavy oil,28 Utah crude oil asphaltene,29 and Maracaibo Lake heavy crude oil in Venezuela.30 These properties are provided at certain temperatures and electromagnetic frequencies. Therefore, some assumptions are required to extrapolate these properties to the temperature and frequency operation range needed in this application. Vrařlstad et al.31 made use of a simple correlation between the permittivity imaginary part and the electromagnetic field frequency σ (T ) + χ ″ (- ) 2πε0 -

σ (T ) 2πε0 -

ε″f at 13.56 MHz

ε″f (×102) at 2.45 GHz

22.23 40 60 80 90

2.88 2.78 2.63 2.42 2.31

0.045 0.108 0.183 0.145 0.116

0.130 0.300 0.481 0.351 0.268

0.720 1.66 2.66 1.94 1.49

10 < T < 100 °C

(16)

Figure 3b demonstrates the effect of the temperature on αf at 2.45 GHz obtained from the substitution of the correlations developed for ε′ and ε″ in eq 4. Such a shape is similar to the one given for ε″ and tan(δ) in ref 23. Dielectric properties are required in this application not only for the transported fluid but also for the pipe materials been considered. Several pipe materials are tested to determine the fluid temperature and velocity profiles. Table 3 illustrates the dielectric properties at 2.45 GHz of the pipe materials taken into account in the simulations. An important condition to take into account for the material selection is the maximum fluid temperature reached at the pipe wall. Transported flammable fluids should be maintained at temperatures in which leaking drops do not ignite. Among the materials shown in Table 3, the silicon carbide permittivity imaginary and real parts at 2.45 GHz reach values of 100 for a wide range of temperatures.33 The substitution of these values in eq 4 provides αp = 234.4 m−1, which is about 500 times larger than the crude oil absorption. Such a large attenuation factor for the pipe material permits delivering most of the generated electromagnetic energy to the fluid portion in contact with the pipe. Such a situation may reduce the fluid viscosity at the pipe wall at the risk of reaching temperatures above the regulation limits for crude oil transportation.

(13)

(14)

Such an approximation is valid for crude oil with a significant dielectric conductivity and permits the calculation of ε″ at different frequencies. The dependency of σ upon the temperature is established by considering the results provided for the Russkoye oil.23 The first three columns of Table 2 provide the data extracted from such a reference. Notice that ε′ monotonically decreases with the temperature, but the loss factor tan(δ) reaches a peak around 63 °C. The last two columns of Table 2 give ε″ = ε′ tan(δ) at 13.56 MHz and 2.45 GHz frequencies. A simple correlation was developed to represent the variation of ε″f with the temperature at a frequency of 2.45 GHz. This approximation consists of a piecewise linear function

ε″f = a + bT

tan(δf)

ε′ = 4.095 − 0.00975T ,

where χ″ is the imaginary part of the complex susceptibility and σ is the dielectric conductivity. Cao et al.32 have suggested that eq 13 can be approximate to the following relation: ε ″ (- , T ) ≈

ε′f

where [a, b] = [−4.125, 0.514] × 10−3 for 10 < T < 63 °C and [a, b] = [57.16, −0.47] × 10−3 for 64 ≤ T ≤ 100 °C. Figure 3a illustrates the approximation to the data in Table 2. Notice that the use of such an approximation is restricted to temperatures between 10 and 100 °C. The results shown for the Alaska heavy crude oil28 illustrate that the permittivity real part does not change with the frequency for - > 100 MHz. The Russkoye crude oil permittivity real part also shows the tendency to remain at 3.9 for - > 60 MHz and 20 °C. Furthermore, ε′ decreases linearly to 80% of its original value when the temperature increases to 100 °C.23 Therefore, the following correlation is used to calculate ε′ as a function of the temperature at 2.45 GHz:

μ = 10h(API)T g(API)

ε ″ (- , T ) =

T (°C)

■

SIMULATION CASES AND RESULTS Lambert’s law is used to relate the electric field amplitude at the pipe wet wall (ER) to the electric field amplitude applied at the pipe dry surface (ER0). E R = E R 0e−αp(R 0− R)

(17)

Substitution of this expression in eq 3 provides the relation between the power absorbed by the fluid per unit volume and the pipe dielectric properties qḟ (r ) = q0̇ ε″f e−2αp(R 0− R)e−2αf (R − r),

(15) 4430

0≤r≤R

(18)


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Figure 3. Effect of the temperature on the Russkoye crude oil dielectric properties at 2.45 GHz.

Table 3. Dielectric Properties of Pipe Materials21,33 material

ε′p

ε″p

tan(δp)

αp (m−1)

ER0 (V/m)

alumina polyethylene silicon carbide lime glass

8.9 2.3 100 6.0

0.009 0.001 100 1.20

0.00010 0.0004 1 0.20

0.0777 0.017 234.4 12.6

180 180.1 321.7 180.4

where q̇0 is the power generated by the electromagnetic field ER0 at free space conditions q0̇ = 2π -ε0E R2 0

(19)

and the subindex p indicates pipe material properties. As explained in the Fluid Model Equations section, two different cases are considered for crude oil transportation under electromagnetic heating. These cases refer to transparent electromagnetic and dielectric pipe materials. Transparent Electromagnetic Pipe. Transparent electromagnetic materials have a negligible attenuation factor α and do not absorb the energy from the electromagnetic field. Substitution of αp = 0 in previous expressions demonstrates that ER = ER0 and qḟ (r ) = q0̇ ε″f e−2αf (R − r),

0≤r≤R

Figure 4. Bulk, center, and wall temperature profiles along the EH for a transparent electromagnetic pipe. The dimensionless temperature θ is defined in the Appendix.

center of the pipe at the outlet flow is about θw − θc = 0.25, which is equivalent to 12.5 °C. The fact that the wall temperature is greater than the center temperature seems to contradict the results obtained for electromagnetic heating in previous publications,35,21 where the core temperature was greater than the wall temperature. Nevertheless, the main difference between the case presented here and previous work is the laminar flow velocity profile of the warmed material. Figure 5 demonstrates the axial velocity profile at the heater outlet. As expected, the flow at the wall is null and the flow at the center doubles the mean velocity U̅ . Such a difference allows for the inner portion of the fluid flow to have lower temperatures than the outer portion. Lambert’s law assumption for electromagnetic power absorption also contributes to the shape of the temperature profile in Figure 4. Figure 6 illustrates the effect of the electromagnetic field amplitude on the pressure gradient. As expected, the larger the electromagnetic field, the lower the pressure drop at the heater outlet. The initial heating portion appears as a straight line; however, it asymptotically flattens as the pressure drop approaches 4.0. Such a decrease in the pressure gradient is expected to significantly reduce the pumping cost for heavy crude oil transportation. Electromagnetic Heated Pipe. A pipe of external radius R0 made of a dielectric material, such as silicon carbide, will

(20)

The pipe is considered externally insulated during electromagnetic heating, which indicates that all of the heat absorbed by the fluid is retained. An expression that provides the dimensionless form of the heat absorbed by the crude oil flow is provided in the Appendix q*̇ f (η) = P*

ε″f −2Rαf (1 − η) e , ε″e

0≤η≤1

where ε″e represents the fluid permittivity imaginary part evaluated at entrance conditions. The increase in the fluid bulk temperature for heavy crude oil transportation is on the order of 50 °C.34 Therefore, the electromagnetic field required for Tbo − Te = 50 °C along a 10 m heater is of 180 V/m. The variable Tbo represents the outlet fluid bulk temperature and is used to define the temperature dimensionless form. Figure 4 illustrates the bulk, center, and wall dimensionless temperature profiles for the Rosskoye crude oil undergoing electromagnetic heating at ER = 180 V/m along the 10 m pipe stretch. Notice that the radial temperature gradient between the wall and 4431


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Substitution of the previous expression into the definition of q* provides the following external boundary condition in dimensionless form q0̇ R2 ∂θ |η = 1 = ke(To − Te) ∂η

∫1

η0

ε″p e−2αpR(η0 − η)ηdη

This boundary condition for constant pipe dielectric properties is reduced to q0̇ R2ε″p (β − 1) ∂θ [1 − e−β(η0 − 1)] |η = 1 = ke(To − Te) β 2 ∂η

where β ≡ 2αpR. In the case of a silicon carbide pipe, for which β > 200, the following approximation is made: ε″ p W0R2 ⎛ 1 − e−β(η0 − 1) ⎞ ∂θ ⎜⎜ ⎟⎟ |η = 1 ≈ ∂η ε″e ke(To − Te) ⎝ β ⎠

Figure 5. Axial velocity profile (U* ≡ U/U̅ ) at the heater outlet for a transparent electromagnetic pipe.

where W0 represents the external electromagnetic power. This type of expression has been used to determine the electromagnetic power absorbed by cylindrical shells during microwave food heating.35 However, in this particular application, eq 22 is used to provide the external boundary condition to the crude oil flow along the EH. Notice that the ratio between the permittivity imaginary parts of the pipe material and entrance fluid (ε″p/ε″e) indicates that the crude oil could be significantly heated at the pipe wall when ε″p ≫ ε″e. Table 3 demonstrates dielectric properties of pipe materials common in industrial applications. Materials with large electromagnetic energy absorption (large tan(δ)) tend to attenuate significantly the electromagnetic field inside the pipe. For this reason, the electromagnetic field needs to be adjusted for electromagnetic absorbent materials to obtain a reasonable temperature increase along the EH. The last column of Table 3 provides the required ER0 to obtain a bulk temperature increase of about 50 °C for different pipe materials. Such an adjustment in the magnetic field induces larger power losses for materials, such as silicon carbide. Nevertheless, heating an external insulated pipe reduces the fluid viscosity in the vicinity of the pipe walls to a level that pressure builds up instead of dropping. Because flow accelerates in the vicinity of the pipe wall, the velocity profile slows at the pipe center. Figure 7 illustrates the pressure gradient profiles for the different materials listed in Table 3. The profiles for alumina, polyethylene, and glass are similar to the one obtained for electromagnetic transparent materials, because of their weak dielectric properties. However, silicon carbide (SiC) shows a different pressure gradient profile because a significant amount of energy is transferred to the portion of fluid in contact with the pipe. The simulations considered for SiC pipes made use of two electromagnetic field levels. The increase of the magnetic field from 180 V/m (nominal) to 321.7 V/m is to provide the latter with the same outlet bulk temperature and pressure gradient than the other materials considered in Figure 7. The bulk temperature at the outlet θbo is defined by

Figure 6. Pressure gradient profile along the heater for electromagnetic fields of 160, 180, and 200 V/m.

warm up because of the electromagnetic field applied at the pipe surface R0. The expression that provides the absorbed heat per unit volume of pipe material is given by qṗ (r ) = q0̇ ε″p e−2αp(R 0− r),

R < r ≤ R0

and the pipe steady-state energy balance is given by eq 2, where all velocities (U and V) are set to zero. 1 ∂ ⎛ ∂Tp ⎞ ⎜rk p ⎟ + qṗ = 0 r ∂r ⎝ ∂r ⎠

(21)

This work focuses on the fluid flow conditions and pressure drop profile for crude oil transportation. Therefore, the calculation of the temperature gradient at the fluid boundary condition is of great interest to define the fluid thermal boundary conditions. Rearrangement of eq 21 demonstrates that, at steady-state conditions and for thermally insulated pipes, all of the electromagnetic energy absorbed by the pipe is transferred to the fluid at r = R. Therefore ⎡ ∂Tp ⎤ = 2π q = ⎢2πrk p ⎥ ∂r ⎦ ⎣ r=R

∫R

R0

(22)

θ bo ≡

Tbo − Te =1 To − Te

where Tbo is the outlet bulk temperature. Such an increase in the electromagnetic field may induce greater energy losses, which might be detrimental for the success of this technology.

qṗ r dr 4432


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about 2000 miles that requires 39 pump stations, which gives an average of L = 50 miles between pump stations. Although KP makes use of fluid emulsions to reduce the crude oil viscosity, this work bases the viscosity and dielectric property calculations on the Russkoye crude oil. The operating temperature range during winter time should be between 100 °C at the pump discharge and 20 °C at the heater suction to minimize the number of pump stations. An average external temperature of Ta = 0 °C is considered during winter time. Figure 8 demonstrates a cylindrical differential section of fluid outside the EH exposed to the external temperature Ta.

Figure 7. Pressure drop profile along the EH for different pipe materials. Profiles of SiC are provided for 180 V/m (nominal) and 321.7 V/m.

In the case of SiC pipes, the localized energy transference at the surface reduces the viscosity in the vicinity of the pipe wall, where the boundary condition U*(1) = 0 holds. The combination of such a boundary condition with a significant drop in the fluid viscosity at the pipe wall gives

Figure 8. Cylindrical differential section of fluid exposed to the external temperature, Ta.

∂ ⎛⎜ ∂U ⎞⎟ < 0 for a developed entrance profile. After a short heating length of less than 2 m, the velocity profile adjusts accordingly and the heating propagates to the core flow. These two effects make the pressure gradient profile of SiC pipes at 321.7 V/m match the alumina, polyethylene, and glass profiles, as illustrated in Figure 7. The next section includes the results obtained for the fluid bulk temperature at the EH outlet conditions to economically determine the potential cost reduction when using EH to transport heavy crude oil along a significant pipe stretch.

U̅ πR2ρCpdT = 2πR[Ta − T (Y )]hdY

(23)

where the overall equivalent heat-transfer coefficient h is assumed constant along the pipe stretch. The solution of the differential eq 23 provides the following temperature profile:

■

ECONOMIC STUDY ANALYSIS The economic study presented in this work only accounts for pipeline design optimization based on the pipeline stretches and location of pump stations. Operating costs related to EH and pump energy consumption cost when compared to fireheater fuel and utility flow costs are not included in this research. The main reason is the variability of utility costs depending upon the region where the pipeline is installed. This paper is intended to provide a general scope of fixed cost reduction without looking at operation costs, which depend upon the heating source and equipment heat-transfer efficiency. The calculation of the velocity, pressure gradient, and temperature profiles inside the EH provides insight about the effect of electromagnetic energy heating on cylindrical fluid flows with high viscosity temperature dependence. Nevertheless, because of the small size of the heater compared to the pipeline length, the economic study of this technology is based on the fluid outlet conditions and the heater efficiency. Figure 2 illustrates that the addition of EH between pump stations will increase their spacing to the point that fewer pump stations might be required. An example of a long pipeline for crude oil transportation is the phase 1 of the Keystone Pipeline (KP) with a total length of

T (y) = Ta + (Td − Ta)e−(2h / UR̅ ρCp)y

(24)

where Td represents the temperature at the pump discharge and y is the distance measured from the pump discharge. Substitution of the parameters given in Table 1 and considering that most of the pipeline is covered by 0.5 in. of a polymer foam with a thermal conductivity of 0.03 W m−1 K−1, we obtain T (y) = e−γy × 100 −2

(25) −1

where γ = 3.2 × 10 miles and the temperature is in Celsius. Therefore, for a 50 mile pipeline stretch measured from the pump discharge at station A to the heater inlet at station B, we obtain T(50) = 20.2 °C, which is within the operating range of the Russkoye crude oil pumping conditions. The calculation of the pressure drop along the 50 mile stretch depends upon the effect of the temperature on the fluid viscosity. Laminar and developed flow assumptions along the 50 mile pipeline section simplify the calculation of the overall pressure drop between pump stations. On the basis of these assumptions, eq 1 is reduced to 1 ∂ ⎛⎜ ∂U ⎞ dP rμ(T ) ⎟ = ⎝ r ∂r dy ∂r ⎠ 4433

(26)


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Article

and for developed flows with a constant density, the continuity eq 6 results in the following parabolic profile:

denoted by yh, is obtained by maximizing the total distance between pump stations (L) while keeping the temperature profile as well as the overall pressure drop within the normal operating range. Hence, the optimal temperature profile that maximizes the pipeline length is given by the solution of the following problem:

⎡ ⎛ r ⎞2 ⎤ U (r ) = 2U̅ ⎢1 − ⎜ ⎟ ⎥ ⎝R⎠ ⎦ ⎣

Substitution of the previous expression into eq 26 provides the following expression for the pipeline pressure drop between pump stations: 8U̅ R2 where μ is in cP ΔP(y) = −

∫0

max L subject to

y

μ(Y )dY

(27)

20 ≤ T (y) ≤ 100 °C |ΔP(L)| ≤ ΔPmax

μ(y) = 4.08(32 + 180e−0.032y)−3.23 × 109

∫0

y

μ(Y )dY

for

0≤y≤L

where ΔPmax = 936.3 kPa

The presence of the EH results in a significant temperature increase ΔTh at yh. On the basis of the optimization variable yh, the pipeline temperature profile is divided into three parts: before, at, and after the EH

and Y is expressed in miles. Therefore, the pressure drop as a function of the pipe length is given by the following expression: ΔP(y) = −12.8

(29)

yh

(28)

T (y) = 100e−γy

where y refers to the pipe total length in miles and the resulting pressure drop ΔP is in kPa. Figure 9 demonstrates the

y − yh

T (y) = T (yh − 0.5l) + ΔTh

l

T (y) = 100e−γ(y − yh − 0.5l)

0 ≤ y < yh − 0.5l yh − 0.5 ≤ y < yh + 0.5l yh + 0.5l ≤ y ≤ L

where l represents the EH length. On the basis of the bulk temperature profile of Figure 4, an EH of length l = 10 m provides an increase of ΔTh = 50 °C in the fluid temperature. Although such a result depends upon the entrance temperature at the EH (Te was set to 20 °C in Figure 4), it is convenient to consider this constant temperature increase for any entrance temperature. Such an assumption allows us to superpose and apply the same EH effect to any pipeline location. The results in Figure 9 can be used to assist the optimizer by reducing the search region toward an optimal yh. In this regards, T(22) = 50 °C, which indicates that yh ≥ 22 miles to satisfy T(y) ≤ 100 °C at the EH outlet. The inspection of the list of constraints also suggests that the inequality |ΔP(L)| ≤ ΔPmax is the one limiting the objective function because ΔP monotonically decreases with y. Therefore, such an inequality constraint is made active; i.e., |ΔP(L)| = ΔPmax = 936.3 kPa. Figure 10 illustrates how the location of the EH affects the distance between pump stations. It is important to highlight that these results maintain the same temperature and pressure discharge conditions as the 50 mile pipeline with no EH

Figure 9. Temperature, viscosity, and pressure drop profiles for the Russkoye crude oil along a 50 mile pipeline.

temperature, viscosity, and pressure drop profiles along the 50 mile pipeline stretch between pump stations. Notice the abrupt increase in viscosity and pressure drop as the crude oil cools from 100 to 20 °C. The total pressure drop along the 50 mile pipeline is of ΔPmax = 936.3 kPa. The first economic question that arises when using an intermediate EH is the additional distance between pump stations that such a device will allow while keeping the pressure drop below ΔPmax. Therefore, the optimal location of the EH,

Figure 10. Effect of the EH location yh on the total length of the pipeline, L. The optimum location of yh = 36.1 miles allows for a total pipe length of L = 73.4 miles before a new pump station is required to raise the fluid pressure. 4434


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installed. The maximum distance of L = 73.4 miles occurs when the heater is placed at yh = 36.1 miles. This result indicates an additional distance of 23.4 miles between pump stations. Temperature, viscosity, and pressure drop profiles for the optimum location of the EH are shown in Figure 11. Notice that the temperature increases to 81.3 °C at the outlet of the EH, and the pressure drop of 936.3 kPa is reached at the pipeline discharge.

is shown that materials with a high heat of absorption, such as SiC, attenuate the electromagnetic field inside the pipe and reduce the effectiveness of the EH. In the case of SiC, the electromagnetic field had to be increased from 180 to 321 V/m to have the same fluid warming effect as other materials with properties similar to transparent electromagnetic materials. An economic analysis of the number and location of EH was made to determine the potential use of these heaters between crude oil pump stations for long pipelines transporting heavy crude oil. The results show that the pipeline length can be increased by 30% when using a transparent pipe material with an EH placed in the optimal location between pump stations. These results were obtained using the Russkoye heavy crude oil because its complete electromagnetic properties were available in the literature. The viscosity of such a crude oil is highly dependent upon the temperature, which justifies high-temperature operations along the pipeline. This work is not intended to provide a detailed feasibility study for the use of EH to reduce the number of pump stations along a large pipeline stretch. Fixed and operational costs of such equipment might make this technology less attractive than the conventional configuration of having heaters only at the pump suction. Nevertheless, the physical effect of inserting EHs permits larger separation between pump stations, which can significantly reduce the heavy crude oil transportation cost because of the high viscosity of such types of fluids at room temperature. Moreover, the use of EHs does not perturb the flow course and requires less equipment infrastructure when compared to furnace heaters. A more detailed study of this technology and the analysis of the operational efficiency of EHs will be performed in the near future.

■

APPENDIX The following dimensionless variables are used to convert the set of partial differential equations in their dimensionless forms: x r U V , η = , U* = , V * = z= RRe R U̅ U̅

Figure 11. Temperature, viscosity, and pressure drop profiles for the optimum location of the EH at yh = 36.1 miles. This arrangement permitted the pipeline length to increase from 50 to 73.4 miles.

ρ* =

The potential distance increase of 23.4 miles between pump stations because of EH permits the reduction of the number of pump stations by 30%. In the hypothetical case of a 2000 mile pipeline length, equivalent to the phase 1 of the KP between Canada and the United States, only 27 pump stations would have been needed instead of 39. This result indicates 12 fewer pump stations than what are actually installed if thermal viscosity reduction would have been considered for such a project. Nevertheless, an economic study that weighs the additional cost of 27 EHs installed in exchange of saving 12 pump stations should be performed to determine the profitability of the proposed approach. Fixed and operational costs of EH and of heavy crude oil pump stations are outside the scope of this work.

Re =

ρ , ρe

P* =

2Rρe U̅ μe

,

P , ρe U̅ 2

Pr =

Ceμe ke

μ* =

,

μ , μe

k* =

Pe = Re*Pr ,

k ke C* =

C Ce

where U̅ represents the uniform entrance velocity used to calculate the constant mass flow rate.

ṁ = ρe U̅ πR2 In the case of electromagnetic heating, the dimensionless temperature is given by θ (z , η) =

■

T (z , η) − Te To − Te

where (To − Te) is a temperature increase reference along the electromagnetic heating section. For numerical convenience, (To − Te) = 50 °C, which is equivalent to the bulk temperature increase along a 10 m EH. The dimensionless heat of absorption is accordingly determined

CONCLUSION The modeling and simulation results of an EH for crude oil transportation were presented in this work. The crude oil dielectric properties and viscosity were adjusted on the basis of the fluid temperature and electromagnetic source frequency. Pipe materials with different electromagnetic properties were tested to determine their effect in the flow pressure gradient. It

q*̇ f (η) ≡ 4435

qḟ (η)R ρe UC ̅ e(To − Te)

(30)


Energy & Fuels

Article

y = axial coordinate along the pipeline z = dimensionless axial coordinate

where qḟ (η) = q0̇ ε″f e−2R[αp(η0 − 1) + αf (1 − η)],

0≤η≤1

Subscripts

For transparent electromagnetic pipes, αp = 0, which makes qḟ (η) = q0̇ ε″f e−2αf R(1 − η),

a = ambient b = bulk d = discharge e = entrance f = fluid h = heater o = outlet p = pipe w = wall 0 = external

0≤η≤1

The electromagnetic fields can be also expressed in terms of the electromagnetic power per unit volume.27 In general, the following relations are used to simplify the expressions applied to calculate the fluid-absorbed heat qḟ (η) = P

ε″f −2Rαf (1 − η) e , ε″e

0≤η≤1

Superscripts

where

* = dimensionless

P = P0e

−2Rαp(η0 − 1)

and P0 = q0̇ ε″e

Greek Letters

Substitution into eq 30 provides a similar dimensionless expression for the fluid-absorbed heat q*̇ f (η) = P*

ε″f −2Rαf (1 − η) e , ε″e

0≤η≤1

where P* =

PR ρe UC ̅ e(To − Te)

■

A heating−pumping station is expected to increase the fluid bulk temperature in 50 °C. Substitution of this information and the electromagnetic properties of the fluid at entrance conditions from Table 1 suggests that an electromagnetic field of ER = 180 V is required for this type of application.

■

α = thermal diffusivity η = dimensionless radial coordinate ε0 = vacuum permittivity ε′ = relative dielectric constant ε″ = electromagnetic heat conversion λ = free space wavelength θ = dimensionless temperature μ = viscosity ρ = density

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

NOMENCLATURE A = pipe cross-section area C = specific heat at constant pressure D = pipe diameter E = amplitude of the electromagnetic field f( ) and g( ) = polynomial functions for viscosity correlation - = frequency k = thermal conductivity l = length of the EH L = length of the pipeline ṁ = mass flow Pe = Peclet number Pr = Prandtl number P = pressure r = radial coordinate Re = Reynolds number R = pipe radius T = temperature U = axial velocity U̅ = average axial velocity V = radial velocity W = electromagnetic power x = axial coordinate along the EH 4436


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Study of Heavy Crude Oil Flows in Pipelines with Electromagnetic

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